The second order question i have been through comes out with lambda +or- 2j and +or- j which is then obvious how to continue.

Could somone point me in the right direction or to where ive gone wrong?

I have an exam on wednesday and realy need to be able to work with any combination of question they give me, It could be first or second order.
I didnt think it mattered which kind i was until differentiating the general solution part.

You're setting up the wrong characteristic equation. Your system looks like this:

where

The solution to the system is

We now have to make sense of the exponential. You use the series definition:

It's the usual series expansion of the exponential function. We can easily find the nth power of if we can find an invertible such that where is diagonal. For then

Taking the nth power of a diagonal matrix is the same as taking the nth power of the diagonal elements, component-wise. Solving is the diagonalization problem. To diagonalize, assuming it's possible to do this, you find the eigenvalues and eigenvectors. The eigenvalues make up the diagonal of , and the eigenvectors make up the columns of . Hence the importance of the eigenvalues for a system of ODE's.

Now, to find the eigenvalues, you have to set not That's where you first went wrong. So carry that correction through, and see what you come up with.

I dont understand why there is a 1 in the first vector bracket? i only get either 6 and -7 or 7 and -6.

Any nonzero scalar multiple of an eigenvector is an eigenvector. Why is that, might you ask? Well, recall that a vector is an eigenvector of the matrix if and only if and for some scalar which is called the eigenvalue corresponding to . So, suppose that is an eigenvector of the matrix with corresponding eigenvalue . Let be a nonzero scalar. Then Since and (since and ), it follows by definition that is an eigenvector of matrix .

Now, if you look at my and multiply by , you might get a more recognizable eigenvector. Does that make sense?

I can see that D and e^Dt are from the two eigenvalues; so do you then work the determinant for that out?

I don't compute the determinant, I just compute the exponential

I'm not sure I understand what you're not understanding. The solution to this problem: pepper me with questions. Anything at all you don't understand, ask about. Ok?